Consider a fair lottery on an infinite domain. We have an infinite collection of tickets, from which one winning ticket t1 is selected at random (and then put back in the collection). Then you buy a random ticket t2. What is the probability that t1=t2, i.e., that you have bought the winning ticket?

Classical probability theory has well-known difficulties in modelling fair infinite lotteries. The obvious suggestion that the probability of you having bought the winning ticket is infinitely small (because the collection of tickets is infinite and the lottery is fair) but non-zero (because it is not impossible that you have bought the winning ticket). But this suggestion is incompatible with classical probability theory because classical probability functions take their values in the real numbers between 0 and 1, and none of these numbers are infinitely small but non-zero.

In this talk I will explore the viability of mathematically developing a non-classical probability theory that allows assigning infinitely small probability values (infinitesimal probability values) to events, and therefore does vindicate the obvious suggestion for answering the main question concerning fair infinite lotteries. I will also discuss some philosophical objections to such infinitesimal probability theories.